( 2, 5, -9 ) + ( -32.034, 94.79, 201.062 ) + ( -2, -5, 9 ) = 

               ( -32.034, 94.79, 201.062 ) + ( -2, -5, 9 ) + ( 2, 5, -9 ) = 
                                             ----------------------------

               ( -32.034, 94.79, 201.062 ) + ( 0, 0, 0)  = 

               ( -32.034, 94.79, 201.062 )

Matrix Addition is Associative

When I give such a problem on a midterm, it is fun to watch students furiously add up the first two matrices, then furiously add in the third matrix.

Some students spoil my fun by realizing that (since matrix addition is commutative) the matrices can be rearranged into a more favorable order. Once the matrices are in a nice order, you can pick whichever "+" you want to do first. That last idea has a name:

Matrix addition is associative. This means that ( a + b ) + c = a + ( b + c ).

This says "first add a to b then add that result to c." The result will be the same as if you did "add a to the result of adding b with c." This works for both row and column matrices of all dimensions.

( 25.1, -19.6 ) + ( -5.0, 9.0 ) + ( 12.4, 8.92 ) + ( -20.1, 10.6 )  =